Is it possible to twist a circle into a sphere?
Yes.

There are probably multiple ways of doing this, but one way is to use a circle with a simple deform modifier set to twist, and using an empty as the twist origin:



By animating the rotation of both the circle and the empty, you can make it twist around, and by animating the deform angle, you can control the amount of twisting.

Result: (click for better quality)



Example .blend